Three-dimensional probabilistic inversion of time-domain airborne electromagnetic (AEM) data is limited by the cost of the forward solve. Even though one simulation takes only tens of seconds, a Bayesian inversion of a survey of millions of soundings requires of order 1010 forward evaluations. To address this, we develop a continually learning neural-operator surrogate of the three-dimensional AEM forward operator that replaces the solver inside the Bayesian inversion. We start from the point of view that regardless of what geological prior is specified, Maxwell's laws remain invariant. Secondly, we avoid the limitation of learning on a single prior by continual learning on consecutive priors, which means our surrogate becomes richer as it is applied in future case studies, either by the authors, or by the scientific community. We use a validity check built on ensemble disagreement to divert cases with measurements outside the training range to the solver. Driven by the surrogate, the identical Markov chain Monte Carlo sampler reproduces the full-solver posterior, and its credible intervals cover the truth within 2.6 percentage points. Applied to the 2013 Capricorn TEMPEST survey in Western Australia, the surrogate inverts over two million soundings in seconds, a computation infeasible for the solver. Testing the geological prior against the entire survey costs minutes. The framework delivers uncertainty-quantified conductivity imaging at survey scale, which we believe is essential to perform near real-time mineral-systems targeting with geophysics.
Figures & tables
Figure 1: The Capricorn TEMPEST survey. (a) The 191 flight lines, colored by the gate time at which each sounding’s decay reaches the noise level. Line 1005801 is marked in black, and the inset locates the survey in Australia. (b) Decay section of line 1005801, log10∣Bz∣ (fT) over the fifteen gates along the 178 km line. Dashed vertical lines mark the three soundings of panel (c). (c) Decays of those three soundings, spanning slow (conductive), median, and fast (resistive) responses. The dashed gray curve is the noise model, and gates below it are shaded and drawn open.
Figure 2: Conceptual geology of the five regions, on one shared log10σ scale. (a) Capricorn, (b) Seward Peninsula, (c) Denmark, (d) Zeeland, and (e) northeastern Wisconsin.
Figure 3: Two realizations of each geological prior, sampled from a Gaussian process. Each row is one prior, and each realization is shown as the three-dimensional volume and a cut-away along a horizontal and a vertical mid-plane.
Prior
Type
Variable
U[a,b]
ℓh (m)
ℓv (m)
α
ν
Source
Capricorn
Conductivity
σregolith
[−2.5,0]
[300, 900]
[120, 300]
[1.0, 2.5]
[2, 5]
Thorne and Tyler (1996) Anand and Paine (2002)
σsediment
[−3.0,−0.1]
σbasement
[−4.0,−1.5]
Geometry
hregolith
[0,120]
dbasement
[40,400]
Seward
Conductivity
σhost
[−3.3,−2.6]
[300, 900]
[120, 300]
[1.0, 3.0]
[1.5, 4]
Emond et al. (2024)
Table 1: Gaussian process hyperparameters of the five geological priors. Each entry is a uniform distribution U[a,b] sampled per realization. σ is the unit conductivity in log10 S/m, h is a layer thickness, d is a top or interface depth, and w is a body width, all in meters. ℓh and ℓv are the horizontal and vertical correlation lengths in meters, α is the horizontal-to-vertical anisotropy factor, and ν is the Matérn smoothness, where a larger ν gives a smoother field. (The Zeeland source is a frequency-domain survey, and only its conductivity structure is used.)
Figure 4: The neural operator surrogate and the sampler it drives. (a) The model encoder, with the eight lowest modes boxed. (b) The prior embedding, the query network and the output. The layered-earth baseline is dashed because the deployed checkpoints set it to zero. (c) The sampler. The gray fill marks the same operator, trained in (a) and (b) and called in (c).
Figure 5: Forward accuracy on the Capricorn test data. (a) Surrogate against forward solver at the fifteen gates, with the 1:1 line. (b) The median sounding, surrogate dashed and forward solver solid. The shaded strip marks gates whose median decay falls below the noise model.
Figure 6: Test error against training set size on Capricorn. Markers and bars are the mean and standard deviation of ten runs at each size, scored on the same 125 test soundings. The line is a power law fitted to all fifty runs.
Figure 7: The validity check on the four regions the operator has not learned, trained on the Capricorn prior alone. (a) The disagreement score of Equation 17 against dissimilarity, the median distance from a test model to the nearest Capricorn training model in a standardized 64-number lateral profile. Markers are the median over soundings and bars run from the tenth to the ninetieth percentile. The dashed line is the threshold u95 , the 95th percentile of the score on the Capricorn validation data. (b) The fraction of each region’s test soundings above the threshold, which are inverted with the forward solver.
Figure 8: Continual learning with and without replay of the earlier priors, across the five geological priors and four arrival orders. Each row is one order, the order our surrogate was trained in first, and each column is a position in the sequence, with the region named in its panel. Arrows run from the stage that taught the region to the end of the sequence.
Figure 9: Sampler-level equivalence on the Capricorn prior. The identical PPM-MCMC sampler runs with the forward solver and with the surrogate, at the same budget and seeds, on one measured sounding of line 1005801. Bands are the 10–90% posterior range of the volume-mean resistivity, lines the medians, pooled over thirty-two chains per operator. The quoted 96% is the median over depth of the surrogate range divided by the solver range.
Figure 10: The falsification test over the whole survey, before and after one turn of the prior revision. Each flown line is colored by its coverage, the fraction of its soundings falling inside the 95% envelope of the prior’s forward responses, and a line is falsified below the 90% level marked on the bar. (a) The deployed Capricorn prior, with the line the updated prior was built from marked in amber. (b) The updated prior.
Figure 11: Whole-survey inversion of Capricorn TEMPEST. The top row is the posterior-median resistivity and the bottom row the width of the posterior, at 38 m (a, c) and 238 m (b, d). The width is the factor in resistivity between the 10th and 90th percentiles, so a value of 2 fixes the resistivity to within a factor of two. All 2,131,667 inverted soundings are shown, plotted one in four for legibility. Panel (a) marks line 1005801, which grounds the prior, and lines 1015101 and 1013902, which it does not explain.
prior
model space (decades)
lines explained (of 190)
lines lost
deployed Capricorn
7.90
127
–
width control
10.53
139
19
regional, independent layers
11.75
176
4
two-component mixture
8.65
184
0
Table 2: The four priors compared. Model space is the width of the prior in decades of log10σ summed over layers, for the mixture the weighted mean of its two components. Lines explained and lines lost are counted over the 190 unseen flight lines.
Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.
Bipin Tiwari, Omer San
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville,2026 TN 37996, USA
Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.
Christian Jimenez-Beltran, Aretha L. Teckentrup, Antonio Vergari +1
School of Mathematics and Maxwell Institute for Mathematical Sciences University of Edinburgh · School of Informatics University of Edinburgh
We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.
Hojjat Kaveh, Ricardo Baptista, Andrew M. Stuart
California Institute of Technology · University of Toronto